Problem: Let $p=x^2+6$. Which equation is equivalent to $(x^2+6)^2-21=4x^2+24$ in terms of $p$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $p^2+4p-45=0$ (Choice B) B $p^2-4p-45=0$ (Choice C) C $p^2+4p-21=0$ (Choice D) D $p^2-4p-21=0$
Solution: We are asked to rewrite the equation in terms of $p$, where ${p}={x^2+6}$. In order to do this, we need to find all of the places where the expression ${x^2+6}$ shows up in the equation, and then substitute ${p}$ wherever we see them! For instance, note that $4x^2+24=4({x^2+6})$. This means that we can rewrite the equation as: $(x^2+6)^2-21=4x^2+24$ $({x^2+6})^2-21=4({x^2+6})$ [What if I don't see this factorization?] Now we can substitute ${p}={x^2+6}$ : $({p})^2-21=4({p})$ Finally, let's manipulate this expression so that it shares the same form as the answer choices: ${p}^2-4{p}-21=0$ In conclusion, $p^2-4p-21=0$ is equivalent to the given equation when $p=x^2+6$.